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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Generalizations of $ {\rm QF}-3$ algebras

Authors: R. R. Colby and E. A. Rutter
Journal: Trans. Amer. Math. Soc. 153 (1971), 371-386
MSC: Primary 16.40
MathSciNet review: 0269686
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Abstract: This paper consists of three parts. The first is devoted to investigating the equivalence and left-right symmetry of several conditions known to characterize finite dimensional algebras which have a unique minimal faithful representation-- QF-$ 3$ algebras--in the class of left perfect rings. It is shown that the following conditions are equivalent and imply their right-hand analog: $ R$ contains a faithful $ \Sigma $-injective left ideal, $ R$ contains a faithful $ II$-projective injective left ideal; the injective hulls of projective left $ R$-modules are projective, and the projective covers of injective left $ R$-modules are injective. Moreover, these rings are shown to be semi-primary and to include all left perfect rings with faithful injective left and right ideals.

The second section is concerned with the endomorphism ring of a projective module over a hereditary or semihereditary ring. More specifically we consider the question of when such an endomorphism ring is hereditary or semihereditary.

In the third section we establish the equivalence of a number of conditions similar to those considered in the first section for the class of hereditary rings and obtain a structure theorem for this class of hereditary rings. The rings considered are shown to be isomorphic to finite direct sums of complete blocked triangular matrix rings each over a division ring.

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Keywords: QF-$ 3$ ring, left perfect ring, unique minimal faithful module, faithful $ \Sigma $-injective left ideal, faithful $ II$-projective injective left ideal, injective hull projective, projective cover injective duality, projective module, endomorphism ring, hereditary ring, semihereditary ring
Article copyright: © Copyright 1971 American Mathematical Society